Assume that the Sharpe$-$Lintner CAPM holds, so the mean$-$variance efficient frontier consists of combinations of Treasury bills and the market portfolio. Nonetheless, some households make the mistake of holding undiversified portfolios that contain only one stock or a few stocks.

$($a$)$ Show that the Sharpe ratio of any portfolio divided by the Sharpe ratio of the market portfolio equals the correlation of that portfolio with the market portfolio.

$($b$)$ Suppose the market is made up of identical stocks, each of which has the same market capitalization, the same mean and variance of return, and the same correlation $\backslash$rho $>0,$ with every other individual stock. Consider the limit as the number of stocks in the market increases. What is the Sharpe ratio of an equally$-$weighted portfolio that contains N stocks divided by the Sharpe ratio of the market portfolio? Interpret.